The Monge Problem for Distance Cost in Geodesic Spaces

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and d (L) is a geodesic Borel distance which makes (X, d (L) ) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem pi which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by d (L) . It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting d (L) -cyclical monotonicity is not sufficient for optimality.